Theorem retancld

Description: Closure of the tangent function. (Contributed by Mario Carneiro, 29-May-2016)

	Ref	Expression
Hypotheses	resincld.1	$$⊢\phi \to A\in ℝ$$
	retancld.2	$$⊢\phi \to \cos A\ne 0$$
Assertion	retancld	$$⊢\phi \to \tan A\in ℝ$$

Proof

Step	Hyp	Ref	Expression
1		resincld.1	$$⊢\phi \to A\in ℝ$$
2		retancld.2	$$⊢\phi \to \cos A\ne 0$$
3		retancl	$$⊢\left(A\in ℝ\wedge \cos A\ne 0\right)\to \tan A\in ℝ$$
4	1 2 3	syl2anc	$$⊢\phi \to \tan A\in ℝ$$